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nature, so complex, and involves the estimation of so many causes, that, to avoid inextricable difficulties, it is necessary to reject some quantities, as being small in comparison of the rest, and to reason as if they had no existence. Dr Stewart, too, it must be confessed, had an additional motive for wishing to simplify his investigation. This was,
his resolution to employ in it no other method than the geometrical, which, however excellent in other respects, is inferior to algebra, for the conducting of very complicated reasonings. The skill of this most profound and experienced geometer could not remedy that defect; and he was reduced to the necessity of rejecting quantities, which were considerable enough to have a great effect on the last result. An error was thereby introduced, which, had it not been for certain compensations, would have become immediately obvious, by giving the sun's distance near three times as great as that which has been mentioned.
The author of the pamphlet, referred to above, was the first who remarked the dangerous nature of these simplifications, and who attempted to estimate the error to which they had given rise. In this last, however, he has not completely succeeded ; and that, too, by committing a mistake similar to that which he censured in Dr Stewart, and by rejecting quantities not less than some which he retained. He observed, however, what produced the
compensation that has been taken notice of, viz. the immense variation of the sun's distance, which corresponds to a very small variation of the motion of the moon's apogee. It is doubtful, whether Dr Stewart was fully apprized of this circumstance; because the geometrical method, elegant and beautiful as it is, rarely presents a general view of the relations, which the magnitudes it treats of bear to one another; and many of these relations may, therefore, escape the most profound geometer, which an algebraist, of more ordinary abilities, would not have failed to discover.
There are other of this author's strictures which we cannot admit as just, but which we will not attempt here either to enumerate or refute. Yet it were doing great injustice to his remarks, not to acknowledge, that, besides being just in the points already mentioned, they are, every where, ingenious, and written with much modesty and good temper. The author, who concealed his name, and permits it now, for the first time, to be publicly mentioned, was Mr Dawson, a surgeon at Sudbury in Yorkshire; a man, as it should seem, who might have enjoyed more of the fame, had he been less satisfied with the possession of knowledge.
A second attack was soon after this made on the Sun's Distance, by Mr Landen; but by no means with the same good temper which has been remarked in the former. He fancied to himself errors in
Dr Stewart's investigation, which have no existence; he exaggerated those that were real, and seemed to triumph in the discovery of them with unbecoming exultation. If there are any subjects on which men may be expected to reason dispassionately, they are certainly the properties of number and extension; and whatever pretexts moralists or divines may have for abusing one another, mathematicians can lay claim to no such indulgence. The asperity of Mr Landen's animadversions must not, therefore, pass uncensured, though it be united with sound reasoning and accurate discussion. The error into which Dr Stewart had fallen, though before taken notice of by Mr Dawson, was first exactly determined in the work before us.* But Mr Landen, in the zeal of correction, brings many other charges against Dr Stewart, the greater part of which seem to have no good foundation. Such are his objections to the second part of the investigation, where Dr Stewart finds the relation between the disturbing force of the sun, and the motion of the apsides of the lunar orbit. For this part, instead of being liable to objection, is deserving of the
It is but justice to remark, that Mr Landen had probably never seen Mr Dawson's Propositions at the time his own were published, the whole impression of them, almost, having been burnt by a fire which consumed the warehouse where they were lodged.
greatest praise, since it resolves, by geometry alone, a problem which had eluded the efforts of some of the ablest mathematicians, even when they availed themselves of the utmost resources of the integral calculus. Sir Isaac Newton, though he assumed the disturbing force very near the truth, computed the motion of the apsides from thence only at one half of what it amounts to in reality; and so, had he been required, like Dr Stewart, to invert the problem, he would have committed an error, not merely of a few thousandth parts, as the latter is alleged to have done, but would have brought out a result double of the truth. * Machin and Callendrini, when commenting on this part of the Principia, found a like inconsistency between their theory and observation. Three other celebrated mathematicians, Clairault, D'Alembert, and Euler, separately experienced the same difficulties, and were led into an error of the same magnitude. It is true, that, on resuming their computations, they found, that they had not carried their approximations to a sufficient length, which, when they had at last accomplished, their results agreed exactly with observation. Mr Walmsley and Dr Stewart were, I think, the first mathematicians who, employing in the solution of this difficult problem, the one the algebraic calculus, and the other the geo
*Prin. Math. Lib. 3. Prop. 3.
metrical method, were led immediately to the truth; a circumstance so much for the honour of both, that it ought, by no means, to be forgotten. It was the business of an impartial critic, while he examined our author's reasonings, to have remarked, and to have weighed these considerations.
We may add, that the accurate measurement of the sun's distance, and the complete theory of the moon's motions, with which science has been enriched, since the time to which we now refer, sufficiently vindicate the principle of Dr Stewart's investigation, and show how much reason he had to expect, that the former might be inferred from the latter with considerable exactness. M. Mayer, from one of the lunar irregularities, computes the sun's parallax to be 7".8, nearly a mean between the parallax already mentioned, and that which has been deduced from the transit of Venus in 1769. *
On the whole, therefore, while it must be acknowledged, that Dr Stewart's determination of the sun's distance is, by no means, free from error, it may safely be asserted, that it contains a great deal which will always interest geometers, and always be admired by them. Few errors in science are redeemed by the display of so much ingenuity, and what is more singular, of so much sound reasoning. The investigation is every where elegant,
*Theoria Lunæ, Sect. 51.